An investigation of statistical thinking in two different contexts: Detecting a signal in a noisy process and determining a typical value

The study describes students’ patterns of thinking for statistical problems set in two different contexts. Fifteen students representing a wide range of experiences with high school mathematics participated in problem-solving clinical interview sessions. At one point during the interviews, each solved a problem that involved determining the typical value within a set of incomes. At another point, they solved a problem set in a signal-versus-noise context [Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259-289]. Several patterns of thinking emerged in the responses to each task. In responding to the two tasks, some students attempted to incorporate formal measures, while others used informal estimating strategies. The different types of thinking employed in using formal measures and informal estimates are described. The types of thinking exhibited in the signal-versus-noise context are then compared against those in the typical value context. Students displayed varying amounts of attention to both data and context in formulating responses to both problems. Suggestions for teachers in regard to helping students attend to both data and context when analyzing statistical data are given.     

Special values of Drinfeld modular forms and algebraic independence

Let A be a polynomial ring in one variable over a finite field and k be its fraction field. Let f be a Drinfeld modular form of nonzero weight for a congruence subgroup of GL₂(A) so that the coefficients of the q _∞-expansion of f are algebraic over k. We consider n CM points α ₁, . . . , α _n on the Drinfeld upper half plane for which the quadratic fields k(α ₁), . . . , k(α _n ) are pairwise distinct. Suppose that f is non-vanishing at these n points. Then we prove that f(α ₁), . . . , f(α _n ) are algebraically independent over k.     

Representation of numbers by quadratic forms in algebraic number fields

Let K be the totally real field of algebraic numbers of degree n=[k2e] with the discriminant D=D(K); t=t(x₁, ..., x_s) a totally positive quadratic form of the determinant d>0 over the ring of integers from the field K; S4. Let be the number of representations over of the number m by the form a complete singular series. It is proved that for given s and n, there exists a constant c such that for N(d)>0 it is not true that for all m with m totally positive.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 68–77, 1986.     

Linear forms,algebraic cycles,and derivatives of L-series

In this note, we state some refinements of conjectures of Gan-Gross-Prasad and Kudla concerning the central derivatives of L-series and special cycles on Shimura varieties. The analogues of our formulation for special values of L-series are written in terms of invariant linear forms on autormorphic representations defined by integrations of matrix coefficients.

     

A note concerning reduced algebraic numbers and equivalence of binary forms

In an earlier paper (J. Wolfskill, J. Number Theory 16 (1983), 205–211) a method was given to determine whether or not two binary forms are equivalent by examining the corresponding problem for the roots. The latter problem was solved, in the case that each form has a real root, by applying the theory of reduced algebraic numbers. However, there are two important problems remaining to be resolved concerning the effectiveness of this method. The first is to show that a given real algebraic number reduces in a bounded (effectively computable) number of steps. The second is to determine under what conditions a form of even degree can be equivalent to its negative. These two problems are solved in this paper.     

Existence and optimality of w-non-adjacent forms with an algebraic integer base

We consider digit expansions in lattices with endomorphisms acting as base. We focus on the w-non-adjacent form (w-NAF), where each block of w consecutive digits contains at most one non-zero digit. We prove that for sufficiently large w and an expanding endomorphism, there is a suitable digit set such that each lattice element has an expansion as a w-NAF. If the eigenvalues of the endomorphism are large enough and w is sufficiently large, then the w-NAF is shown to minimise the weight among all possible expansions of the same lattice element using the same digit system.     

Calculus in European classrooms: curriculum and teaching in different educational and cultural contexts

In this paper we provide a first overview of the landscape with respect to calculus teaching in European classrooms, an area where research is very limited. In particular through a small expert-based survey and a literature review, we trace the development of calculus teaching at schools in a number of European countries and identify commonalities and differences. In the current curriculum developments, we notice a reduction in the content of calculus and a more informal approach. The use of digital tools has started to be integrated in calculus teaching in most countries. However in some nations, teaching of calculus in the classroom is rather traditional, focusing on procedural aspects of knowledge. Moreover, in cases where more informal and conceptual teaching approaches are used in the classroom, often contradictions seem to exist with other contextual matters such as examination requirements. Finally, we discuss the future of calculus teaching in Europe.     

On periods of CUSP forms and algebraic cycles forU(3)

In this paper we discuss relations between the following types of conditions on a representationπ in a cuspidalL-packet ofU(3): (1)L(s, π×ξ) has a pole ats=1 for someξ; (2) aperiod ofπ over some algebraic cycle inU(3) (coming from a unitary group in two variables) is non-zero; and (3) π is atheta-series lifting from some unitary group in two variables. As an application of our analysis, we show that the algebraic cycles on theU(3) Shimura variety arenot spanned (over the Hecke algebra) by the modular and Shimura curves coming from unitary subgroups. All three authors are supported by a grant from the U.S.-Israel Binational Science Foundation; the second author is also supported by an NSF Grant.